Translation-Invariant Free-Fermion-Solvable Spin Models

We give an efficient algorithm for recognizing a free-fermion solution in translation-invariant spin models using graph theory. When such a mapping does exist, the associated free-fermion model may not itself be translation-invariant, but describes fermions coupled to a static gauge field whose configuration is captured by the orientation of a graph. This correspondence allows us to analyze the low-energy behavior of these models using the so-called skew-energy of oriented graphs. We numerically investigate our characterization as applied to a variety of translation-invariant free-fermion models. We expect these results to inform the design of free-fermion-solvable subsystem codes with favorable spectra for error suppression.